Srojiški vesik - Joural of Mechaical Egieerig 63(07)3, 6-70 Received for review: 06-07-06 07 Joural of Mechaical Egieerig. All righs reserved. Received revised for: 07-0-05 DOI:0.5545/sv-je.06.3859 Origial Scieific Paper Acceped for publicaio: 07-0-3 Dyaic Model ad Aalysis of Noliear Vibraio Characerisic of a Curve-Face Gear Drive Cai, Z. Li, C. Zhiqi Cai Chao Li* Chogqig Uiversiy, The Sae Key Laboraory of Mechaical Trasissio, Chia To sudy he oliear dyaic characerisics of curve-face gear drive, a geeralized oliear dyaic odel based o Lagrage Bodgraphs was preseed. The odel cosidered he effecs of he basic paraeers of curve-face gear, such as eshig frequecy, eccericiy k ad he order uber of o-cylidrical gears. Uilizig his siulaio odel, he vibraio characerisics of curve-face gears were aalysed. I is idicaed ha he vibraio respose of he curve-face gear is ore coplex ha ha of he face gear due o he ifluece of exeral exciaio. These characerisics deerie ha his gear pair cao be applied direcly o a rasissio syse wih high speed. Fially, a relaed experie was proposed o prese he validaio of he uerical odel. Keywords: curve-face gear, vibraio respose, liied speed, Lagrage bod-graphs, ie-varyig characerisics Highlighs A oliear dyaic odel based o a Lagrage bod-graph was preseed. The proposed ehod o oly applies o he dyaic aalysis of curve-face gears bu also ca be applied o face gears. The research resuls show ha rb ad R are he ai facor affecig he vibraio respose of curve-face gear, ad reducig he eccericiy k ad raisig he order will iprove he dyaic characerisics. Due o he ifluece of exeral exciaio, which will exacerbae he vibraio, he liied eshig frequecy of curve-face gears is lower ha face gears. These characerisics deerie ha he gear cao be applied o a rasissio syse wih high speed. 0 INTRODUCTION As a ew ype of face gear, curve-face gear cobies he coo characerisics of o-cylidrical gear, bevel gear, ad face gear []. I ca be used i egieerig achiery, agriculural achiery, exile achiery ad high-powered spacecraf, ec. [], ad will be developed owards high speeds ad heavy loads [3]. Therefore, he research o he dyaic characerisics of curve-face gear has icreasigly becoe proie. The research of he oliear vibraio characerisics of gears aily focuses o he bevel gear, spur gear, face gear ad plaeary gear drive; he aalyical approach of he syse geerally uses fiie elee ehod (FEM), ad he Ruge-Kua ehod is applied o solve he differeial equaios of he syse. For he siulaio ad vibraio-based codiio oiorig of a geared syse, a syse wih a appropriae uber of degrees of freedo (DOFs) was odelled [4]. A ivesigaio of he perforace of saisical faul deecio idicaors for hree differe series of crack propagaio scearios was preseed [5]. Fro he vibraio sigal, he ubers of eeh o all gears, he calculaio of ooh esh frequecies ad roaioal speeds of all shafs were deeried [6]. The dyaic behaviour of a sigle-sage spur gear reducer i a rasie regie was sudied [7]. However, for he roary syse wih ass eccericiy, he Lagrage equaio wih geeralized coordiaes expressio is over FEM, as i oly relaes o he kieic eergy ad he poeial eergy of he syse. Therefore, for he echaical syse coposed of paricles ad rigid bodies, he syse equaio ca be obaied quickly by usig he Lagrage equaio. However, due o he differeial calculaio of Lagrage fucio L, is siulaio process is coplex. To overcoe he disadvaage of his odellig ehod, a ew odellig ehod cobied Lagrage equaio ad bod-graph is proposed. Whe he geeralized coordiaes, he paraeer equaio, he force ipu, ad he velociy coversio arix are deeried, he aheaical siulaio of he syse ca be obaied. Copared wih he classical ehod, he odellig process of he Lagrage bod-graph is ore regular, ad he solvig process is ore efficie [8]. Recely, few sudies were available for he curve-face gear drive. Due o he ie-varyig characerisics of o-circular gears, he vibraio respose of his ype of gear is uch coplex ha ha of oral gears. For geeral gears, he exciaios of he vibraio respose are aily ie-varyig esh siffess, ooh rasissio error, ad eshig ipac, ec. The dyaic odel of a spur gear pair, cosiderig he backlash, ie-varyig siffess, *Corr. Auhor s Address: The Sae Key Laboraory of Mechaical Trasissio, Chogqig Uiversiy, Chogqig, Chia, lichao@cqu.edu.c 6
Srojiški vesik - Joural of Mechaical Egieerig 63(07)3, 6-70 ad saic rasissio error, was esablished [9]. Usig sadard ehods for oliear syses, he dyaics of gear syses wih various fauls i eshig siffess was exaied [0]. Cosiderig ie-varyig esh siffess, esh dapig, ooh error, ooh fricio, backlash ad bearig clearace, a bedig-orsio coupled oliear dyaic odel of a face-gear drive syse was esablished []. A oliear dyaic odel of a face gear syse was preseed by he coceraed paraeers, gear clearaces, rasissio error, eshig siffess, eshig dapig, brace siffess, suppor dapig ad exciig frequecy []. However, for a ocircular gear, i addiio o he above facors, he regular exeral exciaio, caused by he basic paraeers of he o-circular gear, such as eshig frequecy, eccericiy k ad he order of he ocylidrical gear, is oe of he ai ifluecig facors. For he vibraio respose of he o-circular gear, a dyaic odel of a o-cylidrical gear based o he Lagrage bod-graphs heory was esablished [3]. A virual experieal odel of he o-cylidrical gear was esablished, ad he correspodig dyaic respose was obaied [4]. Cosiderig he ievaryig eshig siffess, a esioal vibraio odel of a o-circular gear was developed [5]. Bearig i id ipu shaf ad oupu shaf exciaio, he ie-varyig eshig siffess, saic rasissio error, gear backlash, a oliear dyaic odel of he wisig vibraio of he curve-face gear was esablished [6]. However, he syseaic aalysis of he effec of he basic paraeers of curve-face gear o he vibraio respose reai ukow, ad he pheoeo of he liied speed of curve-face gear is o ye aalysed. I his paper, a geeralized oliear dyaic odel of curve-face gear based o Lagrage bodgraph was proposed, ad he relaed vibraio resposes were aalysed usig bod-graph heory. Wih he chage of eshig frequecy, eccericiy k ad he order of o-circular gear, differe vibraio resposes were show i he ie hisory char ad phase diagra. Fially, a relaed experie was proposed o verify he correcess of he heoreical ode. VIBRATION MODEL REVIEW The eshig process of he curve-face gear pair is illusraed as show i Fig.. Coordiaes S (X Y Z ) ad S (X Y Z ) are fixed o he frae of he ocylidrical gear ad he curve-face gear. The ocylidrical gear ad he curve-face gear roae alog heir ow axis O Z ad axis O Y wih roaio agles θ ad θ, respecively. Accordig o he characerisics of he curveface gear, he eshig force ca be decoposed io forces alog he direcio of axis X ad axis Y, ad here is o radial force. Copared wih he bevel gear, he supporig srucure of he curve-face gear pair is relaively siple, ad he DOFs of vibraio are less. Fig.. Modellig of curve-face gear drive The rasissio raio of curve-face gear drive ad ca be expressed as: R i = r ( ). () θ The value of cylidrical radius R ad polar radius r(θ ) of o-cylidrical gear ca be represeed as: kcos( θ ) r ( θ ) = a( k ) R. () π r θ θ π 0 = ( ) d. Mehod of Lagrage Bod-Graph Wih he coplexiy of syse icreases, he coveioal ehod esablishig he oliear vibraio odel is quie edious. Especially for curveface gear drives, he ifluece of ecceric ass us be cosidered [3]. Alhough he odellig process by he Lagrage equaio, based o kieic eergy ad poeial eergy ca be easier, he solvig process reaiig difficul. To overcoe his disadvaage, cobiig wih he ehod of bod-graph, he syse equaio ad he calculaio process ca be easier. A basic Lagrage bod-graph odel is show i Fig., X represes he field source, such as dapig filed - R, Ierial field - I ad elasic field - K, ec. por lik he serial eergy pors. MIF represes he arix rasforaio TX q G is he geeralized velociy vecor which is he firs order derivaive of geeralized displacee q G F X represes he correspodig 6 Cai, Z. Li, C.
Srojiški vesik - Joural of Mechaical Egieerig 63(07)3, 6-70 force, such as he D Aleber force F I, elasic force F K/k, dissipaive force F R/r ad ipu force F s. as: Fig.. Basic Lagrage bod-graph odel V X is he velociy vecor which ca be expressed VX = TX q G. (3) I his paper, a geeralized oliear odel based o he Lagrage bod-graph is proposed as show i Fig. 3. q X Y θ X Y θ, (6) G = [ ] ad he displacee vecor i elasic field ca be defied as: = [ ] qk X Y cx X Y c X, (7) where c = cos a, c = si a. Fig. 4. Dyaic odel of curve-face gear drive syse Fig. 3. Lagrage bod-graph odel of syse I Fig. 3, por liks he serial eergy pors. Each leg (force) has a coribuio i por. F + τ + F + F = F + F + F. (4) I I K R S k r The hidde dyaic equaio of he Lagrage bod-graph odel is represeed as follows: T I T dq I I d dt dq T C dc I T dq C G + W C C W + d d d d + T Kf q T Rg q G I G + K ( )+ ( )= G R T T Ke T R de = S τ S + k e + r e. (5) d G as: Ipu force vecor τ S ca be defied as: τ S [ ] = 0 0 T 0 0 T. (8) Gear coprehesive error vecor ca be defied e = e [... ] 6 (9) For he cylidrical gear drive, he pressure agle is equal o he gear shaper. However, his is o so for he curve-face gear pair. As show i Fig. 5, L as, L as ad L 3 as 3 are he pich curve of he o-cylidrical gear, curveface gear ad gear shaper, respecively. Three ooh profiles G, G ad G 3 are i agecy a poi P.. Derivaio of Dyaic Model The dyaic odel of a curve-face gear drive is show i Fig. 4. Z ad Y are roaioal axes of he o-cylidrical gear ad curve-face gear, respecively. The value of basic paraeers, eshig siffess k i, dapig c i (i = x, x, y, y, ), ad orque T j ( j =, ) are show i Table... Deeriaio of Key Vecors The displacee vecor i geeralized coordiaes ca be represeed as: Fig. 5. Pressure agle of curve-face gear drive Accordig o he defiiio of pressure agle, he pressure agle α u of he gear shaper is cosa. However, he pressure agle α of he curve-face gear Dyaic Model ad Aalysis of Noliear Vibraio Characerisic of a Curve-Face Gear Drive 63
Srojiški vesik - Joural of Mechaical Egieerig 63(07)3, 6-70 refers o he agle bewee absolue velociy v p, ad he oral force F which ca be obaied as: α α π = u+ arca r( θ) r ( θ). (0) Se X as he saic rasissio error. X = c ( X + r θ ) + cy c ( X + Rθ ) c Y. () b However, i he acual codiio, gear backlash b is a ipora oliear facor of gear dyaics, which ca be expressed as: X b X > b f( x) = 0 X b. () X + b X < b.. Deeriaio of Trasfor Marix The curve-face gear drive is a coplex rasissio syse coposed of uliple rigid bodies. The coprehesive velociy rasfor arix i a ierial field ca be expressed as: TV TI =. (3) TW Moreover, velociy rasfor arix i a poeial source field ca be expressed as: T = S E( 6 6 ). (4) Velociy rasfor arix i a dapig field ca be expressed as: T R 0 0 0 0 0 0 0 0 0 0 c cc crb c cc cr =. (5) 0 0 0 0 0 0 0 0 0 0 cc c cr b cc c cr Whe akig i accou he gear coprehesive error, he velociy rasfor arix i a dapig field ca be expressed as: T r = diag( r b R). (6) The velociy rasfor arix i a elasic field ca be expressed as: T K = T. (7) Siilarly, he velociy rasfor arix i a elasic field, cosiderig he gear coprehesive error, ca be defied as: Tk = diag( rb R). (8) The roaio arix ca be expressed as: R cosθ siθ 0 0 0 0 siθ cosθ 0 0 0 0 0 0 0 0 0 C C =, 0 0 0 cosθ siθ 0 0 0 0 siθ cosθ 0 0 0 0 0 0..3 Deeriaio of Boudary Codiios (9) The boudary codiios of curve-face gear drive are represeed as follows: a) Dapig field The dapig arix i a dapig field ca be expressed as: R diag c c c c c c. (0) = ( x y x y ) Siilarly, whe akig io accou gear coprehesive error, i ca be defied as: Re = diag( cx cy c cx cy c ), () where c = cc, c = cc. x y b) Ierial field The ass arix i a ierial field ca be expressed as: M I = 0 I. () 0 C c) Elasic field The siffess arix i a dapig field ca be expressed as: K = diag( k k k k k k ). x y x y (3) Siilarly, whe cosiderig gear coprehesive error, i ca be defied as: Ke = diag( kx ky k kx ky k), (4) where k x = c k, k y = c k...4 Deeriaio of Syse Equaio Subsiuig Eqs. (6) o (4) io Eq. (5), he dyaic oliear differeial equaios ca be expressed as: X + c X x + kx X+ ck δ = cc δ Y + cy y + kyy+ ck δ = cc δ I zzθ + kδrb = T cδ r b X + cx X + k x X ck δ = ccδ, (5) Y + cy Y + ky Y ck δ = cc δ I zzθ kδr= T + cδ R 64 Cai, Z. Li, C.
Srojiški vesik - Joural of Mechaical Egieerig 63(07)3, 6-70 where δ = X e(), δ = X e ( ).. Siffess VIBRATION EXCITATION OF SYSTEM Meshig siffess is coposed of he average siffess k ad ie-varyig siffess. k () = k + k cos( iω ϕ ). (6) i i= The value of k i () is associaed wih coac raio ε ad ca be expressed as: cos( πi( ε )) ki() = k iπ cos( πi( ε )) ϕi = arca si( π i( ε )) kp = k / ε h i p, (7) where ε fis he oogeic haroic fucio, which ca be obaied as: ε( θ)= ε + εicos( iθ σ i ). (8) i= As he siffess of he bearig is highly sesiive o he load o he bearig i he radial direcio, he selecio of he adoped bearig siffess is ecessary. As a kid of o-circular gear, he curve-face gear is aily applied i ediu-low speed ad lowload occasios. Specifically, he chage of load has lile effec o he siffess. I his paper, he bearig siffess is regarded as a cosa, which ca be obaied as: Fr k = li =, (9) Fr 0 + + F + 3. δ l 03. CF δ 0 where he basic paraeers,, C ad F r ca be obaied fro he referece [7].. The Drivig Torque Alhough he drivig orque T is cosa, he drive orque T of he curve-face gear is variable due o he ie-varyig rasissio raio. The relaioship bewee he drivig orque T ad he drive orque T ca be expressed as: r T = ( T I α ) i, (30) where agular acceleraio a ca be described as: r as: dω aω k( k )si( ω ) α = =. (3) d R( kcos( ω )) The equivale exciaio force ca be described F TI zzrb( θ) + IzzRT ( Iα) () =. I R + I r ( θ ) z z zz b 3 DIMENSIONLESS DYNAMIC EQUATION (3) For curve-face gear, he value of r(θ ) varies wih he roaio agle θ. Tha is, he base radius r b of he o-cylidrical gear is o cosa ad he velociy of he saic rasissio error ca be calculaed as: X = ( Xcos a + rb θ+ rb θ) + Ysi a ( X cos a + Rθ ) Y si a. (33) Le ω = k e, τ = ω, ω h = Ω h / ω ; x j = X j / b ; y j = Y j / b ; λ = X / b e(τ); ϑ j = θ j / b ad e(τ) = (e 0 + e r cos(ω h τ+ϕ i )) / b. The, Eq. (5) ca be diesioless as: x + ζ x x+ cζ λ + κx x+ cκ λ = 0 y y c + ζ y + ζλ + κy y + c κ λ = 0 ϑ + rbζh λ + rb κhλ = ξ x + ζ x x cζλ, (34) + κx x cκ = 0 λ y + ζ yy cζλ + κy y c κλ = 0 ϑ Rζhλ Rκhλ = ξ where ω ij = k ij j ; ζ ij = c ij /( j ω ); ζ j = c /( j ω ); κ j = k (τ)/( j ω ); κ hj = k (τ)/( I j ω ); κij = ω ij ω ; ζ hj = c /( I j ω ); (i = x, y, j =, ). 4 SIMULATION AND ANALYSIS As ca be see i Eq. (34), i is a higher-order oliear vibraio syse. I his paper, he bod-graph ehod was iroduced o solve he diesioless dyaic of differeial equaios. Whe he geeralized coordiaes, he paraeer equaio, he force ipu ad he velociy coversio arix (Eqs. (6) o (4)) are deeried, he ie hisory char ad phase diagra ca be obaied, ad he resuls are preseed i he diesioless for. Due o he exeral exciaio of he curve-face gear pair caused by eccericiy k, eshig frequecy Dyaic Model ad Aalysis of Noliear Vibraio Characerisic of a Curve-Face Gear Drive 65
Srojiški vesik - Joural of Mechaical Egieerig 63(07)3, 6-70 ω h ad order of he o-cylidrical gear, a uique dyaic characerisic of curve-face gear syse is produced as show i Figs. 6 o 8. Table. Basic paraeers of curve-face gear drive Paraeers Curve-face gear Module [] 4 Teeh of drivig gear z 8 Teeh of drive gear z 36 Tooh widh B [] 30 Drive orque [N] 700 Weigh of drive gear [kg] 0.55 Weigh of drive gear [kg].673 Roaioal ieria I zz [kg ] 3.5 0 4 Roaioal ieria I zz [kg ] 8. 0 3 Meshig rigidiy k [N ] 8. 0 8 Meshig dapig c [N s ].5 0 3 Suppor siffess k [N ] k x = 7.4 0 8 ; k x =. 0 9 ; k y = 6.48 0 8 ; k y =.3 0 9 Suppor dapig c [N s - ] c x =.6 0 3 ; c x =.6 0 4 ; c y =. 0 3 ; c y =.55 0 4 4. Dyaic Resuls by Eccericiy Eccericiy k is oe of he ai paraeers ha reflecs he vibraio respose of he curve-face gear drive. Fig. 6 gives he resuls of gear vibraio respose i differe eccericiy k. Referrig o ie hisory char (Fig. 6), whe he eshig frequecy is cosa, he uber of vibraio periods reais uchaged wih eccericiy k. Wih he icrease of eccericiy k, he vibraio apliude icreases ad he vibraio period covers fro haroic-periodic o quasi-periodic oio. The ai cause for his pheoeo depeds o: ) Accordig o Eq. (34), r b ad R are he ai facors affecig he vibraio respose of he syse. For he curve-face gear, r b chages accordigly wih he chage of eshig frequecy, which is differe fro he ordiary gear. This leads o a regular flucuaio of he oral load (or ecceric load) uder he sae load orque wih he exeral exciaio chages. ) I addiio o he ifluece of he eshig frequecy, r b ad R are also affeced by eccericiy k ad he order uber of ocylidrical gear. The flucuaig rage of r b icreases bu he value of R decreases wih he rise of eccericiy k, which resuls i he icrease of exeral exciaio ad he ieio of vibraio. a) b) c) Fig. 6. Ifluece of eccericiy o syse respose; ie hisory char ad phase diagra for a) eccericiy k = 0., b) eccericiy k = 0., ad c) eccericiy k = 0.3 As show i he phase diagra (Fig. 6), he equilibriu posiios of he curve-face gear shif backwards ad forwards i he process of vibraio: he equilibriu posiios of vibraio vary periodically wih eshig ie, ad he value of he vibraio apliude rises wih he icrease of eccericiy k. The ai reaso is ha he oral load (or ecceric load) bewee he ooh is affeced by r b ad R. Uder he exciaio of eshig siffess, he vibraio displacee chages ad hus he deviaio of he equilibriu posiio chages correspodigly. I coclusio, he dyaic perforace eds o be beer wih he decrease of eccericiy k. 4. Dyaic Resuls by he Order k As aoher sigifica facor of he vibraio respose of he curve-face gear drive, he differe order uber of o-cylidrical gear deerie differe dyaic perforaces as show i Fig. 7. As show i he ie hisory char (Fig. 7), whe he eshig frequecy is cosa, he uber of vibraio periods reais uchaged wih he order uber of o-cylidrical gears. Wih he icrease of he order uber, he vibraio apliude decreases ad he flucuaio period icreases fro o 3. The ai reasos are as follows: 66 Cai, Z. Li, C.
Srojiški vesik - Joural of Mechaical Egieerig 63(07)3, 6-70 ) The chage period of r b icreases, ad he apliude reais uchaged, which leads o he icrease of he flucuaio period of vibraio; ) The vibraio apliude decreases as he value of R icreases ad he order uber icreases. As show i he phase diagra (Fig. 7), he periodic chage ad ai reaso of he equilibriu posiios are he sae as i Fig. 6. However, he deviaio of he equilibriu posiio reais cosa ad he uber of he equilibriu posiios decreases due o he chagig rules of r b. ) The equilibriu posiios vary periodically wih eshig frequecy sice r b chages accordigly wih he chage of eshig frequecy. For he vibraio respose of curve-face gear, while he eshig frequecy icreases, he chage of value r b affecs he exeral exciaio, ieria force ad cerifugal force, which iesifies he vibraio of he syse. Cosequely, he vibraio perforace of he curve-face gear is ore coplex ha ha of he face gear. I his siuaio, boh sides of he ooh surface will coac aleraively whe he eshig frequecy reaches a cerai value. This is o allowed i he vibraio respose of he gear rasissio syse, which resuls i a liied eshig frequecy of he curve-face gear. a) a) b) b) c) Fig. 7. Ifluece of order o syse respose, ie hisory char, ad phase diagra for: a) order =, b) order = ad c) order = 3 I coclusio, he dyaic perforace eds o be beer wih he icrease of he order. 4.3 Dyaic Resuls by Meshig Frequecy For various eshig frequecies, he dyaic respose of curve-face gear drive shows differe flucuaio apliudes ad equilibriu posiios. As show i Fig. 8, whe he eshig frequecy icreases, eccericiy k ad he order uber are cosa: ) The uber of vibraio periods, he equilibriu posiios ad he vibraio apliude all icrease, which is he sae as he vibraio of he face gear []; c) Fig. 8. Ifluece of eshig frequecy o syse respose; ie hisory char ad phase diagra for dyaic respose whe a) ω h = 0.3, b) ω h = 0.35 ad c) ω h = 0.359 Fig. 9. Dyaic respose uder liied speed; ie hisory char ad phase diagra As show i Fig. 9, whe he eshig frequecy is 0.378, he reverse coac appears. This reveals ha Dyaic Model ad Aalysis of Noliear Vibraio Characerisic of a Curve-Face Gear Drive 67
Srojiški vesik - Joural of Mechaical Egieerig 63(07)3, 6-70 he gear will o work properly sice he eshig frequecy reaches he liied value. The eshig frequecy ca be covered o a liied speed of 790 rp, which idicaes ha he curve-face gear is suiable for ediu ad low speed siuaios. 5. Theory Validaio 5 VALIDATION AND ANALYSIS Whe k = 0, he curve-face gear ca be degeeraed io a ideal face gear. A haroic respose is show by he ie hisory char ad phase diagra (Fig. 0), which is he sae as he exisig coclusio of face gear []. This eas ha he dyaic equaio o oly ca be applied o he dyaic aalysis of curveface gear, bu also o face gear. Dyaic sigals ca be acquired by easurig he vibraio value alog X, Y ad Z direcios o he bearig cap. No. ad 5 are acceleraio sesors; No. is he dyaic sigal acquisiio odule; No. 3 is he Ehere chassis. No. 4 is he siplified curve-face gear pair. Sice he acquisiio process is affeced by he ierferece of rado sigals, acquired daa will be picked by soe exisig errors. To reduce he ifluece of he error o he experieal resuls, he daa us be colleced afer cerai processig. The chagig rules of he vibraio respose alog X ad Y are he sae. Therefore, he variaio resposes alog X i he eshig period were aalysed as show i Fig.. a) Fig. 0. Theory validaio, ie hisory char, ad phase diagra The roo cause lies i he fac ha here is o esseial differece bewee o-circular gears ad coveioal gears excep for he differece of rasissio raio fucio. Thus, he ipac of exeral exciaio o loger exiss. 5. Experie Validaio The experie is based o he rasissio plafor of he curve-face gear as show i Fig.. Fig.. Experie plafor b) c) Fig.. The copariso bewee experieal ad heoreical daa; ie hisory char ad phase diagra for ipu velociy a) 00 rp, b) 400 rp, ad c) 600 rp The velociy sigal (Fig. b) ad displacee sigal (Fig. a) ca be rasfored direcly by he firs ad wice iegral of he acceleraio sigal, separaely. The experieal resuls prove ha: ) whe he eshig frequecy is cosa, he periodic flucuaio of he vibraio respose of he curve-face gear is ore coplex ha ha of he face gear due o he ifluece of he basic paraeers of curve-face gear; ) he copariso bewee experie ad siulaio resuls deosraes ha he aalyical resuls show good agreee wih he relaed experieal daa. I is reasoable o coclude ha he experie preses he validaio of uerical odel. 68 Cai, Z. Li, C.
Srojiški vesik - Joural of Mechaical Egieerig 63(07)3, 6-70 8 NOMENCLATURES Fig. 3. Experieal resul uder liied speed; vibraio acceleraio ad ie hisory char For he validaio of he liied speed of he curve-face gear, he vibraio respose ca be obaied whe he ipu speed is aroud 790 rp, as show i Fig. 3. The vibraio displacee of he gear pair is egaive, which idicaes he reverse coac of he gear pair ad will cause he desrucio of he gear. Thus, he gear pair reaches he lii speed posiio, which represes he validaio of he aheaical odel. 6 CONCLUSIONS ) A oliear dyaic odel based o a Lagrage bod-graph was preseed. The proposed ehod o oly applies o he dyaic aalysis of he curve-face gear, bu also ca be applied o he face gear. ) The exeral exciaio, caused by r b ad R which are affeced by he ecceric k ad he order of o-circular gear, is he ai facor affecig he vibraio respose of curve-face gear. The research resuls show ha reducig he eccericiy ad raisig he order uber will iprove he dyaic characerisics of syse. 3) The vibraio respose of he curve-face gear is also affeced by he eshig frequecy. However, due o he ifluece of exeral exciaio, which will exacerbae he vibraio, he liied eshig frequecy of he curve-face gear is lower ha ha of he face gear. These characerisics deerie ha he gear cao be applied o he rasissio syse wih high speed. 4) The experie resuls are i good agreee wih he heory aalysis of he odels. I is reasoable o coclude ha he odel preseed is capable of efficiely siulaig he vibraio respose of he curve-face gear. 7 ACKNOWLEDGMENT This work suppored by Research Progra suppored by fro he Naioal Naural Sciece Foudaio (575537), Chia ad Graduae Sude Research Iovaio Projec (No. CYB509). R he radius of curve-face gear, [] r(θ ) he polar radius of o-cylidrical gear, [] k eccericiy he order uber of o-cylidrical gear he order uber of curve-face gear a sei ajor axis of ellipse, [] F I D 'Aleber force, [N] τ I ieria force, [N] F K elasic force, [N] F S ipu force, [N] F R dissipaive force, [N] F k elasic force caused by gear coprehesive error, [N] F r dissipaive force caused by gear coprehesive error, [N] T I coprehesive velociy rasfor arix i ierial field q G displacee vecor i geeralized coordiaes syse I ass arix i ierial field T W rasfor arix of agular velociy i ierial field C C roaio arix i ierial field I C ieria arix i ieria field T K velociy rasfor arix i elasic field K siffess arix T R velociy rasfor arix i dapig field R dapig arix T S velociy rasfor arix i poeial source τ S ipu force vecor T k velociy rasfor arix of gear coprehesive error i elasic field K e siffess arix of gear coprehesive error e gear coprehesive error vecor T r velociy rasfor arix of gear coprehesive error i dapig field R e dapig arix of gear coprehesive error T drivig orque, [N ] T drive orque, [N ] α pressure agle of curve-face gear, [ ] r b base radius of o-cylidrical gear, [] T V raslaioal rasfor arix i ierial field T W agular velociy rasfor arix i ierial field E ui vecor k eshig siffess, [N ] k average eshig siffess, [N ] Ω h gear eshig frequecy ε ie-varyig coac raio ε average coac raio σ i iiial phase i-order ie-varyig coac raio ε i Dyaic Model ad Aalysis of Noliear Vibraio Characerisic of a Curve-Face Gear Drive 69
Srojiški vesik - Joural of Mechaical Egieerig 63(07)3, 6-70 ω oial siulaig frequecy b he oral backlash, [] η j oial displacee (agle) j = (x, y, ϑ, λ) τ oial ie e() diesioless coprehesive error ω h diesioless siulaig frequecy 9 REFERENCES [] Li, C., Gog, H., Nie, N., Ze, Q., Zhag, L. (0). Geoery desig, hree-diesioal odelig ad kieaic aalysis of orhogoal flucuaig gear raio face gear drive. Proceedigs of he Isiuio of Mechaical Egieers, Par C: Joural of Mechaical Egieerig Sciece, vol. 7, o. 4, p. 779-793, DOI:0.77/095440645338. [] Livi, F.L., Wag, J.-C., Bossler, R.B., Che, Y.-J., Heah, G., Lewicki, D.G. (994). Applicaio of face-gear drives i helicoper rasissios. Joural of Mechaical Desig, vol. 6, o. 3, p. 67-676, DOI:0.5/.99434. [3] Li, C., Cai, Z. (05). Modelig of dyaic efficiecy of curve-face gear pairs. Proceedigs of he Isiuio of Mechaical Egieers, Par C: Joural of Mechaical Egieerig Sciece, vol. 30, o. 7-8, p. 09-, DOI:0.77/0954406563308. [4] Mohaed, O.D., Raaalo, M., Aidapää, J.-O. (05). Dyaic odellig of a oe-sage spur gear syse ad vibraio-based ooh crack deecio aalysis. Mechaical Syses ad Sigal Processig, vol. 54-55, p. 93-305, DOI:0.06/j.yssp.04.09.00. [5] Mohaed, O.D., Raaalo, M., Aidapää, J.-O., Kuar, U. (03). Vibraio sigal aalysis for gear faul diagosis wih various crack progressio scearios. Mechaical Syses ad Sigal Processig, vol. 4, o. -, p. 76-95, DOI:0.06/j. yssp.03.06.040. [6] Sawalhi, N., Radall, R.B. (04). Gear paraeer ideificaio i a wid urbie gearbox usig vibraio sigals. Mechaical Syses ad Sigal Processig, vol. 4, o. -, p. 368-376, DOI:0.06/j.yssp.03.08.07. [7] Khabou, M.T., Bouchaala, N., Chaari, F., Fakhfakh, T., Haddar, M. (0). Sudy of a spur gear dyaic behavior i rasie regie. Mechaical Syses ad Sigal Processig, vol. 5, o. 8, p. 3089-30, DOI:0.06/j.yssp.0.04.08. [8] Karopp, D. (977). Lagrage s equaios for coplex bod graph syses. Joural of Dyaic Syses, Measuree, ad Corol, vol. 99, o. 4, p. 300-306, DOI:0.5/.3473. [9] She, Y., Yag, S., Liu, X. (006). Noliear dyaics of a spur gear pair wih ie-varyig siffess ad backlash based o icreeal haroic balace ehod. Ieraioal Joural of Mechaical Scieces, vol. 48, o., p. 56-63, DOI:0.06/j.ijecsci.006.06.003. [0] Liak, G., Friswell, M.I. (005). Dyaics of a gear syse wih fauls i eshig siffess. Noliear Dyaics, vol. 4, o. 4, p. 45-4, DOI:0.007/s07-005-398-y. [] Li, T., Ra, X. (0). Noliear vibraio characerisic aalysis of a face-gear drive. Joural of Vibraio ad Shock, vol. 3, o., p. 5-3. (i Chiese) [] Li, X., Zhu,R., Li Z. (03). Aalysis of coupled vibraio of face gear drive wih o-orhogoal. Joural of Ceral Souh Uiversiy (Sciece ad Techology), vol. 44, o. 6, p. 74-80. [3] Wag, A., Ma,Q., Liu, L. (006). Modelig ad Siulaio of Ellipse Gears. Joural of Mechaical Trasissio, vol. 30, o. 4, p. 7-0. (i Chiese) [4] Che,Y., Hu,Q., Luo, L. (009). Research o a ovel rasissio echais based o he priciple of space curve esh. Joural of Mechaical Trasissio, vol. 33, o., p. -4. (i Chiese) [5] Arakawa, A. (995). Coac raio of ocircular gears. Trasacios of he Japa Sociey of Mechaical Egieers Series C, vol. 6, o. 585, p. 093-099, DOI:0.99/ kikaic.6.093. (i Japaese) [6] Li, C., Liu, Y., Gu, S. (04). Aalysis of oliear wisig vibraio characerisics of orhogoal curve-face gear drive. Joural of he Brazilia Sociey of Mechaical Scieces ad Egieerig, vol. 37, o. 5, p. 499-505, DOI:0.007/ s40430-04-096-y. [7] Wu, H., Wag, J., A, Q. (008). Sudy o he calculaig ehod of radial siffess of apered roller bearig. Lubricaio Egieerig, vol. 33, o. 7, p. 39-43. 70 Cai, Z. Li, C.